arXiv subjects

Oriol Reig Fité

Publications and source records attributed to Oriol Reig Fité.

4 recordsLinked to original sources

arXiv2026

Hankel and Multiplication Tensor Completions for Cactus Rank

We show that the Hankel flat extension formulation of the cactus algorithm is equivalent to a completion problem for multiplication tensors of Artinian Gorenstein algebras. The unknown Hankel moments are canonically identified with the undetermined tensor coefficients, and under this identification the symbolic multiplication matrices and their commutation equations coincide. This shows that the usual degree extension formulation is a coordinate realization of a variable extension problem with marked generators. We further use Borel-fixed and squat staircases to reduce the family of candidate basis shapes in the resulting algorithm.

math.AC↗

arXiv2026

Determinantal computation of minimal local GADs

We study local generalized additive decompositions (GADs) of homogeneous polynomials and their associated points schemes through their local inverse systems. We verify that their construction and algebraic properties are independent of the chosen apolarity action. We propose a determinantal method for computing minimal local GADs by minimizing the rank of a symbolic inverse system. When the locus of minimal supports is finite, this procedure provides a practical tool to determine all minimal local decompositions without tensor extensions. We prove that this finiteness is guaranteed whenever the local GAD-rank of the form does not exceed its degree. We analyze both generic and special cases, provide computational evidence assessing the impact of different choices for minors in the determinantal algorithm, and compare our approach with existing algorithms for local apolar schemes.

math.AC↗

arXiv2025

A refinement on the local cactus rank algorithm

We present an algorithm to recover a minimal local apolar scheme to a homogeneous polynomial $F$. The socle degree of the scheme determines whether it is evinced by a Generalized Additive Decomposition (GAD) of $F$ or of an extension. We give constructive procedures for both cases and compute the Hilbert function efficiently via Hankel operators.

math.AG↗

arXiv2025

Computing Path Signature Varieties in Macaulay2

The signature of a path is a non-commutative power series whose coefficients are given by certain iterated integrals over the path coordinates. This series almost uniquely characterizes the path up to translation and reparameterization. Taking only fixed degree parts of these series yields signature tensors. We introduce the Macaulay2 package $\texttt{PathSignatures}$ to simplify the study of these interesting objects for piecewise polynomial paths. It allows for the creation and manipulation of parametrized families of paths and provides methods for computing their signature tensors and their associated algebraic varieties.

math.AG↗